Riemann curvature tensor

About: Riemann curvature tensor is a research topic. Over the lifetime, 6248 publications have been published within this topic receiving 138871 citations. The topic is also known as: Riemann–Christoffel tensor.

Plebanski gravity without the simplicity constraints

Abstract: In Plebanski's self-dual formulation general relativity becomes SO(3) BF theory supplemented with the so-called simplicity (or metricity) constraints for the B-field. The main dynamical equation of the theory states that the curvature of the B-compatible SO(3) connection is self-dual, with the notion of self-duality being defined by the B-field. We describe a theory obtained by dropping the metricity constraints, keeping only the requirement that the curvature of the B-compatible connection is self-dual. It turns out that the theory one obtains is to a very large degree fixed by the Bianchi identities. Moreover, it is still a gravity theory, with just two propagating degrees of freedom as in GR.

Variational Properties of the Gauss-Bonnet Curvatures

Abstract: The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metrics with unit volume are generalized Einstein metrics and once restricted to a pointwise conformal class of metrics are metrics with constant Gauss-Bonnet curvature.

Einstein-Vector Gravity, Emerging Gauge Symmetry and de Sitter Bounce

Abstract: We construct a class of Einstein-vector theories where the vector field couples bilinearly to the curvature polynomials of arbitrary order in such a way that only Riemann tensor rather than its derivative enters the equations of motion. The theories can thus be ghost free. The U(1) gauge symmetry may emerge in the vacuum and also in some weak-field limit. We focus on the two-derivative theory and study a variety of applications. We find that in this theory, the energy-momentum tensor of dark matter provides a position-dependent gauge-violating term to the Maxwell field. We also use the vector as an inflaton and construct cosmological solutions. We find that the expansion can accelerate without a bared cosmological constant, indicating a new candidate for dark energy. Furthermore we obtain exact solutions of de Sitter bounce, generated by the vector which behaves like a Maxwell field in the later time. We also obtain a few new exact black holes that are asymptotic to flat and Lifshitz spacetimes. In addition, we construct exact wormholes, and Randall-Sundrum II domain walls.

Discovering the Kerr and Kerr-Schild metrics

Abstract: The story of this metric begins with a paper by Alexei Zinovievich Petrov (1954) where the simultaneous invariants and canonical forms for the metric and conformal tensor are calculated at a general point in an Einstein space. This paper took a while to be appreciated in the West, probably because the Kazan State University journal was not readily available, but Felix Pirani (1957) used it as the foundation of an article on gravitational radiation theory. He analyzed gravitational shock waves, calculated the possible jumps in the Riemann tensor across the wave fronts, and related these to the Petrov types. I was a graduate student at Cambridge, from 1955 to 1958. In my last year I was invited to attend the relativity seminars at Kings College in London, including one by Felix Pirani on his 1957 paper. At the time I thought that he was stretching when he proposed that radiation was type N, and I said so, a rather stupid thing for a graduate student with no real supervisor to do†. It seemed obvious that a superposition of type N solutions would not itself be type N, and that gravitational waves near a macroscopic body would be of general type, not Type N. Perhaps I did Felix an injustice. His conclusions may have been oversimplified but his paper had some very positive consequences. Andrzej Trautman computed the asymptotic properties of the Weyl tensor for outgoing radiation by generalizing Sommerfeld’s work on electromagnetic radiation, confirming that the far field is Type N. Bondi, M.G.J. van der Burg and Metzner (1962) then introduced appropriate null coor-